When Pi is Not 3.14 | Infinite Series | PBS Digital Studios
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange answerers link it when explaining that pi depends on the metric, and when introducing taxicab and L^p distances and the shape of their circles.
Recommended by 4 people across 4 different months, 2011 to 2018.
What it is
An 11-minute PBS Infinite Series episode, written and hosted by mathematician Kelsey Houston-Edwards, showing how the value of pi changes when distance is measured with different metrics such as the taxicab metric and other L^p metrics. It visualizes circles in these geometries and notes that pi is at least 3 in any such norm.
What people said
Recommends jimjim · 2011-01 · Mathematics Stack Exchange · on “Why is $\pi $ equal to $3.14159...$?”
Update : When Pi is Not 3.14 | Infinite Series | PBS Digital Studios [this video] can be used for teaching. Why is this number equal to 3.14159....? Why is it not some other (ir)rational number? Answer is that with the usual Euclidean metric that is the number that one gets, the value of $\pi$ is dependent on the geometry that is being used, so on a sphere the $\pi$ used to obtain the area will be different. Another question to ask is that if $S=\pi_1 r^2$ is the area of the circle and $C=\pi_2 2 r$ is the circumference why $\pi_1=\pi_2 = \pi$ ? What geometries or metrics will result in $\pi_1 \neq \pi_2$ ?
Recommends Bobson Dugnutt · 2017-01 · Mathematics Stack Exchange · on “types of distances between points”
The term you're looking for is a metric . From the wiki [LINK] : A metric or distance function is a function that defines a distance between each pair of elements of a set. A fun example is the so-called " taxicab metric [LINK] ", or $L^1$, where the distance function (or metric) is given by $$d(\mathbf{x,y})=\sum_{i=1}^n|x_i-y_i|,$$ where $\bf{x,y}$ are $n$-dimensional vectors. $L^2$ is the usual Euclidean metric ( here [LINK] is a link on $L^p$) and there are many others. PBS Infinite Series recently made a nice video [this video] on the subject.
Recommends Andrei · 2017-06 · Mathematics Stack Exchange · on “Geometric interpretation of $x^3+y^3+z^3=k^3$”
The answer it's a sphere, but not with the usual Euclidian metric, but instead is the $L^3$ metric in a 3D space. In the Euclidian or $L^2$ metric, in $n$ dimensions, the norm is given by $$||X||_2=(x_1^2+x_2^2+...)^{1/2}$$ In general, the distance in the $L^p$ metric is given by the above equation by replacing $2$ with $p$ $$||X||_p=(x_1^p+x_2^p+...)^{1/p}$$ You can find more details on wikipedia [LINK] or there is a nice movie [this video] on youtube (the shape of $L^3$ circle is shown at 6 minutes and 17 seconds in this clip).
Recommends user451844 · 2017-07 · Mathematics Stack Exchange · on “What is the mathematical motivation behind metric geometry”
this isn't an answer as I'm not that advanced ( especially in this area) but the taxicab metric in theory could have real world applications: [this video] and in theory if a taxi can only move in certain directions in a city, you can figure out the minimum distance a taxi, would need to drive to get from place to place ( taxicab metric, idealized city) though I've also seen that talked about in graph theory terms.
Cites Andrei · 2018-01 · Mathematics Stack Exchange · on “Multi-metrics in metric space”
See Geometric interpretation of $x^3+y^3+z^3=k^3$ [LINK] , including the links in the reply. Suppose you want to draw a "circle" using $d_p=\sqrt[p]{\sum_i|v_i|^p}$ metric. Obviously for $p=2$ you get your usual circle. As you increase $p$, the circle would transform into a square with the sides parallel to the axes. In the limit $p\rightarrow\infty$ you can take the largest term in the sum outside of the root $p$. What you are left is with $1$ plus terms with absolute value less than $1$, raised to power of $\infty$. Those terms all become $0$. That's why your metric $d_\infty$ is described by the maximum value. For part 2, $S$ is a cube centered on $0$, with side $2$. For part 3, the answe